2 Answers
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No. The only actions with true equivalence in both environments are taps and clicks.

Swipes and drag-and-drops aren't the same, because dragging and dropping is much harder ergonomically, so you wouldn't rely on it as you would with swipes. Swipes don't map to mouse movements either, as you can have nonsemantic mouse movements as the user innocently moves the cursor to hit some other target. Pinches and zooms do map, but on desktop, zooms tend to be managed by the host application or OS rather than the webpage in question.

Even if the actions were comparable, most of the time your mobile and desktop experiences are going to differ because the contexts of use are so different. Users' priorities in the different contexts won't be the same, so the UIs won't be the same either.